Group #34206819 — subgroups

Subgroups — #34206819

Order 20224.

316 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23765166	C₂	1	1	yes	maximal, characteristic
2	#23765005	C₂	1	1	yes	maximal, characteristic
3	#23761486	C₂	1	2	yes	maximal
4	#23761482	C₂	1	2	yes	maximal
5	C₇₉×#2960	C₂	1	1	yes	maximal, characteristic
6	C₂×#14579101	C₂²	1	2	yes	—
7	C₂×#14579101	C₂²	1	2	yes	—
8	#19098384	—	2	2	no	—
9	#19098341	—	2	2	no	—
10	#533×C₇₉	C₄	1	1	yes	characteristic
11	#446×C₇₉	—	2	2	no	—
12	C₈×Q₈×C₇₉	—	2	2	no	—
13	#424×C₇₉	C₄	1	1	yes	characteristic
14	#424×C₇₉	C₄	1	1	yes	characteristic
15	#423×C₇₉	C₄	1	1	yes	characteristic
16	#405×C₇₉	C₂²	1	1	yes	characteristic
17	#405×C₇₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₇₉	C₂²	1	1	yes	characteristic
19	#14579101	—	2	2	no	—
20	#14579101	—	2	2	no	—
21	#14579101	—	2	2	no	—
22	#14579101	Q₈	1	4	yes	complemented
23	C₂×C₇₉⋊C₁₆	—	2	2	no	—
24	C₂×C₇₉⋊C₁₆	—	2	2	no	—
25	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
26	C₂×M₁₆×C₇₉	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈×C₇₉	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₇₉	C₂³	1	1	yes	characteristic
29	C₄⋊Q₈×C₇₉	—	2	1	no	—
30	#125×C₇₉	—	2	1	no	—
31	C₂²⋊Q₈×C₇₉	—	2	2	no	—
32	C₄×Q₈×C₇₉	—	2	2	no	—
33	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
34	#117×C₇₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₇₉	C₂×C₄	1	1	yes	characteristic
36	C₄⋊C₈×C₇₉	—	2	2	no	—
37	C₄⋊C₈×C₇₉	—	2	2	no	—
38	C₄⋊C₈×C₇₉	—	2	2	no	—
39	#97×C₇₉	—	2	1	no	—
40	#97×C₇₉	D₄	1	2	yes	—
41	C₄×C₈×C₇₉	—	2	1	no	—
42	C₄×C₈×C₇₉	—	2	1	no	—
43	C₄×C₈×C₇₉	—	2	1	no	—
44	C₄×C₈×C₇₉	D₄	1	2	yes	—
45	C₇₉⋊C₁₆	—	2	4	no	—
46	C₇₉⋊C₁₆	—	4	2	no	—
47	C₂×Q₈×C₇₉	—	2	2	no	—
48	C₂²×C₄×C₇₉	—	2	1	no	—
49	C₂²×C₄×C₇₉	C₂²×C₄	1	1	yes	characteristic
50	M₁₆×C₇₉	—	2	2	no	—
51	M₁₆×C₇₉	—	2	2	no	—
52	C₂×C₈×C₇₉	—	2	1	no	—
53	C₂×C₈×C₇₉	—	2	1	no	—
54	C₂×C₈×C₇₉	—	2	2	no	—
55	C₂×C₈×C₇₉	C₂²⋊C₄	1	2	yes	—
56	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₇₉	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₇₉	C₂×Q₈	1	2	yes	—
59	C₂×C₈×C₇₉	#41	1	2	yes	—
60	C₄⋊C₄×C₇₉	—	2	4	no	—
61	C₄⋊C₄×C₇₉	—	2	2	no	—
62	C₄⋊C₄×C₇₉	—	2	1	no	—
63	C₄⋊C₄×C₇₉	—	2	1	no	—
64	C₂²⋊C₄×C₇₉	—	2	1	no	—
65	C₂²⋊C₄×C₇₉	M₁₆	1	2	yes	—
66	C₄²×C₇₉	—	2	1	no	—
67	C₄²×C₇₉	—	2	1	no	—
68	C₄²×C₇₉	C₂²⋊C₄	1	2	yes	—
69	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
70	C₄²×C₇₉	M₁₆	1	1	yes	characteristic
71	C₂³×C₇₉	C₂×M₁₆	1	1	yes	characteristic
72	Q₈×C₇₉	—	4	4	no	—
73	C₂×C₄×C₇₉	—	2	1	no	—
74	C₂×C₄×C₇₉	—	2	1	no	—
75	C₂×C₄×C₇₉	—	2	2	no	—
76	C₂×C₄×C₇₉	—	2	2	no	—
77	C₂×C₄×C₇₉	—	4	2	no	—
78	C₂×C₄×C₇₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₇₉	C₄×Q₈	1	2	yes	—
80	C₂×C₄×C₇₉	C₂×M₁₆	1	1	yes	characteristic
81	C₂×C₄×C₇₉	#131	1	2	yes	—
82	C₈×C₇₉	—	2	2	no	—
83	C₈×C₇₉	—	4	1	no	—
84	C₈×C₇₉	—	4	1	no	—
85	C₈×C₇₉	C₂²⋊Q₈	1	2	yes	—
86	C₈×C₇₉	#124	1	1	yes	characteristic
87	C₈×C₇₉	C₄⋊Q₈	1	1	yes	characteristic
88	C₂²×C₇₉	—	4	1	no	—
89	C₂²×C₇₉	#407	1	1	yes	characteristic
90	C₂²×C₇₉	#446	1	2	yes	—
91	C₄×C₇₉	—	2	2	no	—
92	C₄×C₇₉	—	4	1	no	—
93	C₄×C₇₉	—	4	2	no	—
94	C₄×C₇₉	—	4	2	no	—
95	C₄×C₇₉	#391	1	1	yes	characteristic
96	C₄×C₇₉	#408	1	1	yes	characteristic
97	C₄×C₇₉	#446	1	2	yes	—
98	#13270	—	79	1	no	maximal, Sylow
99	C₂×C₇₉	—	2	1	no	—
100	C₂×C₇₉	—	2	2	no	—
101	C₂×C₇₉	#1985	1	1	yes	characteristic
102	#2960	D₇₉	1	1	yes	characteristic
103	#2153	—	79	1	no	—
104	#2110	—	79	1	no	—
105	#1378	—	79	2	no	—
106	#1376	—	79	2	no	—
107	C₇₉	#13270	1	1	yes	characteristic, complemented, Sylow
108	#533	Dic₇₉	1	1	yes	characteristic
109	C₂×M₃₂	—	79	2	no	—
110	C₂×M₃₂	—	79	2	no	—
111	#446	—	2	2	no	—
112	C₈×Q₈	—	2	2	no	—
113	#424	Dic₇₉	1	1	yes	characteristic
114	#424	Dic₇₉	1	1	yes	characteristic
115	#423	Dic₇₉	1	1	yes	characteristic
116	#405	C₂×D₇₉	1	1	yes	characteristic
117	#405	C₂×D₇₉	1	1	yes	characteristic
118	C₂×C₄×C₈	C₂×D₇₉	1	1	yes	characteristic
119	#364	—	158	2	no	—
120	#347	—	158	2	no	—
121	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
122	C₂×M₁₆	C₂×Dic₇₉	1	1	yes	characteristic
123	C₂²×C₈	C₂×Dic₇₉	1	1	yes	characteristic
124	C₂²×C₈	C₂²×D₇₉	1	1	yes	characteristic
125	C₄⋊Q₈	—	2	1	no	—
126	#125	—	2	1	no	—
127	C₂²⋊Q₈	—	2	2	no	—
128	C₄×Q₈	—	2	2	no	—
129	#117	C₂×Dic₇₉	1	1	yes	characteristic
130	#117	C₂×Dic₇₉	1	1	yes	characteristic
131	C₂×C₄²	C₂×Dic₇₉	1	1	yes	characteristic
132	M₃₂	—	79	4	no	—
133	M₃₂	—	158	2	no	—
134	M₃₂	—	158	2	no	—
135	M₃₂	—	158	2	no	—
136	C₂×C₁₆	—	158	2	no	—
137	C₂×C₁₆	—	158	2	no	—
138	C₄⋊C₈	—	2	2	no	—
139	C₄⋊C₈	—	2	2	no	—
140	C₄⋊C₈	—	2	2	no	—
141	#97	—	2	1	no	—
142	#97	C₂²⋊D₇₉	1	2	yes	—
143	C₄×C₈	—	2	1	no	—
144	C₄×C₈	—	2	1	no	—
145	C₄×C₈	—	2	1	no	—
146	C₄×C₈	C₂²⋊D₇₉	1	2	yes	—
147	C₂×Q₈	—	2	2	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂²×Dic₇₉	1	1	yes	characteristic
150	M₁₆	—	2	2	no	—
151	M₁₆	—	2	2	no	—
152	C₂×C₈	—	2	1	no	—
153	C₂×C₈	—	2	1	no	—
154	C₂×C₈	—	2	2	no	—
155	C₂×C₈	C₂²⋊Dic₇₉	1	2	yes	—
156	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
157	C₂×C₈	C₂×C₂²⋊D₇₉	1	1	yes	characteristic
158	C₂×C₈	Q₈×D₇₉	1	2	yes	—
159	C₂×C₈	#11929625	1	2	yes	—
160	C₄⋊C₄	—	2	4	no	—
161	C₄⋊C₄	—	2	2	no	—
162	C₄⋊C₄	—	2	1	no	—
163	C₄⋊C₄	—	2	1	no	—
164	C₂²⋊C₄	—	2	1	no	—
165	C₂²⋊C₄	#11929608	1	2	yes	—
166	C₄²	—	2	1	no	—
167	C₄²	—	2	1	no	—
168	C₄²	C₂²⋊Dic₇₉	1	2	yes	—
169	C₄²	#11929608	1	1	yes	characteristic
170	C₄²	#11929608	1	1	yes	characteristic
171	C₁₆	—	316	2	no	—
172	C₁₆	—	158	4	no	—
173	C₂³	C₂×#11929608	1	1	yes	characteristic
174	Q₈	—	4	4	no	—
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	4	2	no	—
180	C₂×C₄	C₂×C₂²⋊Dic₇₉	1	1	yes	characteristic
181	C₂×C₄	Q₈×Dic₇₉	1	2	yes	—
182	C₂×C₄	C₂×#11929608	1	1	yes	characteristic
183	C₂×C₄	#14579168	1	2	yes	—
184	C₈	—	2	2	no	—
185	C₈	—	4	1	no	—
186	C₈	—	4	1	no	—
187	C₈	#14579137	1	2	yes	—
188	C₈	#14579150	1	1	yes	characteristic
189	C₈	#14579159	1	1	yes	characteristic
190	C₂²	—	4	1	no	—
191	C₂²	#19098527	1	1	yes	characteristic
192	C₂²	#19098694	1	2	yes	—
193	C₄	—	2	2	no	—
194	C₄	—	4	1	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	2	no	—
197	C₄	#19098470	1	1	yes	characteristic
198	C₄	#19098530	1	1	yes	characteristic
199	C₄	#19098694	1	2	yes	—
200	C₂	—	2	1	no	—
201	C₂	—	2	2	no	—
202	C₂	#23764341	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23765166

C₂

yes

maximal, characteristic

#23765005

C₂

yes

maximal, characteristic

#23761486

C₂

yes

maximal

#23761482

C₂

yes

maximal

C₇₉×#2960

C₂

yes

maximal, characteristic

C₂×#14579101

C₂²

yes

—

C₂×#14579101

C₂²

yes

—

#19098384

—

#19098341

—

#533×C₇₉

C₄

yes

characteristic

#446×C₇₉

—

C₈×Q₈×C₇₉

—

#424×C₇₉

C₄

yes

characteristic

#424×C₇₉

C₄

yes

characteristic

#423×C₇₉

C₄

yes

characteristic

#405×C₇₉

C₂²

yes

characteristic

#405×C₇₉

C₂²

yes

characteristic

C₂×C₄×C₈×C₇₉

C₂²

yes

—

—

—

Q₈

yes

complemented

C₂×C₇₉⋊C₁₆

—

C₂×C₇₉⋊C₁₆

—

C₂×M₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂×M₁₆×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂×C₄

yes

characteristic

C₂²×C₈×C₇₉

C₂³

yes

characteristic

C₄⋊Q₈×C₇₉

—

#125×C₇₉

—

C₂²⋊Q₈×C₇₉

—

C₄×Q₈×C₇₉

—

#117×C₇₉

C₂×C₄

yes

characteristic

#117×C₇₉

C₂×C₄

yes

characteristic

C₂×C₄²×C₇₉

C₂×C₄

yes

characteristic

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

C₄⋊C₈×C₇₉

—

#97×C₇₉

—

#97×C₇₉

D₄

yes

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

—

C₄×C₈×C₇₉

D₄

yes

—

C₇₉⋊C₁₆

—

C₇₉⋊C₁₆

—

C₂×Q₈×C₇₉

—

C₂²×C₄×C₇₉

—

C₂²×C₄×C₇₉

C₂²×C₄

yes

characteristic

M₁₆×C₇₉

—

M₁₆×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

—

C₂×C₈×C₇₉

C₂²⋊C₄

yes

—

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×D₄

yes

characteristic

C₂×C₈×C₇₉

C₂×Q₈

yes

—

C₂×C₈×C₇₉

#41

yes

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₄⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

—

C₂²⋊C₄×C₇₉

M₁₆

yes

—

C₄²×C₇₉

—

C₄²×C₇₉

—

C₄²×C₇₉

C₂²⋊C₄

yes

—

C₄²×C₇₉

M₁₆

yes

characteristic

C₄²×C₇₉

M₁₆

yes

characteristic

C₂³×C₇₉

C₂×M₁₆

yes

characteristic

Q₈×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

—

C₂×C₄×C₇₉

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₉

C₄×Q₈

yes

—

C₂×C₄×C₇₉

C₂×M₁₆

yes

characteristic

C₂×C₄×C₇₉

#131

yes

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

—

C₈×C₇₉

C₂²⋊Q₈

yes

—

C₈×C₇₉

#124

yes

characteristic

C₈×C₇₉

C₄⋊Q₈

yes

characteristic

C₂²×C₇₉

—

C₂²×C₇₉

#407

yes

characteristic

C₂²×C₇₉

#446

yes

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

—

C₄×C₇₉

yes

C₄×C₇₉

yes

C₄×C₇₉

yes

—

#13270

—

maximal, Sylow

C₂×C₇₉

—

100

C₂×C₇₉

—

101

C₂×C₇₉

#1985

yes

characteristic

102

#2960

D₇₉

yes

characteristic

103

#2153

—

104

#2110

—

105

#1378

—

106

#1376

—

107

C₇₉

#13270

yes

characteristic, complemented, Sylow

108

#533

Dic₇₉

yes

characteristic

109

C₂×M₃₂

—

110

C₂×M₃₂

—

111

#446

—

112

C₈×Q₈

—

113

#424

Dic₇₉

yes

characteristic

114

#424

Dic₇₉

yes

characteristic

115

#423

Dic₇₉

yes

characteristic

116

#405

C₂×D₇₉

yes

characteristic

117

#405

C₂×D₇₉

yes

characteristic

118

C₂×C₄×C₈

C₂×D₇₉

yes

characteristic

119

#364

—

158

—

120

#347

—

158

—

121

C₂×M₁₆

C₂×Dic₇₉

yes

characteristic

122

C₂×M₁₆

C₂×Dic₇₉

yes

characteristic

123

C₂²×C₈

C₂×Dic₇₉

yes

characteristic

124

C₂²×C₈

C₂²×D₇₉

yes

characteristic

125

C₄⋊Q₈

—

126

#125

—

127

C₂²⋊Q₈

—

128

C₄×Q₈

—

129

#117

C₂×Dic₇₉

yes

characteristic

130

#117

C₂×Dic₇₉

yes

characteristic

131

C₂×C₄²

C₂×Dic₇₉

yes

characteristic

132

M₃₂

—

133

M₃₂

—

158

—

134

M₃₂

—

158

—

135

M₃₂

—

158

—

136

C₂×C₁₆

—

158

—

137

C₂×C₁₆

—

158

—

138

C₄⋊C₈

—

139

C₄⋊C₈

—

140

C₄⋊C₈

—

141

#97

—

142

#97

C₂²⋊D₇₉

yes

—

143

C₄×C₈

—

144

C₄×C₈

—

145

C₄×C₈

—

146

C₄×C₈

C₂²⋊D₇₉

yes

—

147

C₂×Q₈

—

148

C₂²×C₄

—

149

C₂²×C₄

C₂²×Dic₇₉

yes

characteristic

150

M₁₆

—

151

M₁₆

—

152

C₂×C₈

—

153

C₂×C₈

—

154

C₂×C₈

—

155

C₂×C₈

C₂²⋊Dic₇₉

yes

—

156

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

157

C₂×C₈

C₂×C₂²⋊D₇₉

yes

characteristic

158

C₂×C₈

Q₈×D₇₉

yes

—

159

C₂×C₈

#11929625

yes

—

160

C₄⋊C₄

—

161

C₄⋊C₄

—

162

C₄⋊C₄

—

163

C₄⋊C₄

—

164

C₂²⋊C₄

—

165

C₂²⋊C₄

#11929608

yes

—

166

C₄²

—

167

C₄²

—

168

C₄²

C₂²⋊Dic₇₉

yes

—

169

C₄²

#11929608

yes

characteristic

170

C₄²

#11929608

yes

characteristic

171

C₁₆

—

316

—

172

C₁₆

—

158

—

173

C₂³

C₂×#11929608

yes

characteristic

174

Q₈

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₂×C₄

—

178

C₂×C₄

—

179

C₂×C₄

—

180

C₂×C₄

C₂×C₂²⋊Dic₇₉

yes

characteristic

181

C₂×C₄

Q₈×Dic₇₉

yes

—

182

C₂×C₄

C₂×#11929608

yes

characteristic

183

C₂×C₄

#14579168

yes

—

184

C₈

—

185

C₈

—

186

C₈

—

187

C₈

#14579137

yes

—

188

C₈

#14579150

yes

characteristic

189

C₈

#14579159

yes

characteristic

190

C₂²

—

191

C₂²

#19098527

yes

characteristic

192

C₂²

#19098694

yes

—

193

C₄

—

194

C₄

—

195

C₄

—

196

C₄

—

197

C₄

#19098470

yes

characteristic

198

C₄

#19098530

yes

characteristic

199

C₄

#19098694

yes

—

200

C₂

—

201

C₂

—

202

C₂

#23764341

yes

characteristic

	subgroup	G/H
center	C₈	#14579150
commutator	C₂²×C₄×C₇₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₉
Fitting	C₇₉×#2960	C₂
socle	C₂×C₇₉	#1985
prime radical	C₂³×C₇₉	C₂×M₁₆
cosocle	C₂²×C₈×C₇₉	C₂³
hypercenter	#2960	D₇₉
hypocenter	C₇₉	#13270
LCS	G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2960

subgroup

G/H

center

C₈

#14579150

commutator

C₂²×C₄×C₇₉

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₉

Fitting

C₇₉×#2960

C₂

socle

C₂×C₇₉

#1985

prime radical

C₂³×C₇₉

C₂×M₁₆

cosocle

C₂²×C₈×C₇₉

C₂³

D₇₉

C₇₉

G ▷ C₂²×C₄×C₇₉ ▷ C₂×C₇₉ ▷ C₇₉

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2960

N	H ≅ G/N	type
22. #14579101	Q₈	semidirect
107. C₇₉	#13270	semidirect

H ≅ G/N

type

22. #14579101

Q₈

semidirect

107. C₇₉

#13270

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#23765166	2
#23765005	2
#23761486	2
#23761482	2
C₇₉×#2960	2
#13270	79

subgroup

index

C₇₉×#2960

Subgroups — #34206819

Conjugacy classes of subgroups

Complements

Maximal subgroups